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        riddles >> putnam exam (pure math) >> Compare an Integral and a Number
        
(Message started by: Barukh on Aug 11^th, 2004, 10:12am)

Title: Compare an Integral and a Number
Post by Barukh on Aug 11^th, 2004, 10:12am

Which number is bigger?

1. [int]₀^[pi] e^(sin²x) dx

2. 3[pi]/2

Title: Re: Compare an Integral and a Number
Post by Aryabhatta on Aug 11^th, 2004, 11:34am

let f(x) = e^sin[sup2]x
Now f(x) [ge] 1 + sin[sup2]x + (sin⁴x)/2

So
[int]₀^[pi]f(x)dx > [int]₀^[pi](1 + sin[sup2]x)dx = 3[pi]/2

as [int]₀^[pi](sin⁴x)/2 > 0 and

[int]₀^[pi]sin[sup2]xdx = 2[int]₀^[pi]/2sin[sup2]xdx = 2[int]₀^[pi]/2cos[sup2]xdx = [int]₀^[pi]/2(sin[sup2]x + cos[sup2]x)dx = [pi]/2